Goodness of fit test or independence (contingency table) test — χ² statistic, p-value, and effect size.
Test whether observed frequencies match expected frequencies.
Test whether two categorical variables are independent. Enter the contingency table — one row per line, values separated by commas.
The chi-square (χ²) test is a statistical test for categorical data. It compares observed counts to expected counts to determine whether a significant difference exists.
There are two main variants: the goodness-of-fit test (one variable, comparing to a theoretical distribution) and the test of independence (two variables, testing if they are associated).
Goodness of fit: You have one categorical variable and want to test whether the observed frequencies match an expected distribution. Example: "Are the outcomes of a die roll uniformly distributed?"
Test of independence: You have a contingency table with two categorical variables and want to know if they are associated. Example: "Is gender associated with product preference?" Enter the raw cell counts as a matrix.
Cramér's V is an effect size measure for chi-square tests, ranging from 0 (no association) to 1 (perfect association). It corrects for the fact that chi-square grows with sample size, making it a pure measure of effect regardless of n.
Interpretation: small (~0.1), medium (~0.3), large (~0.5). Use Cramér's V alongside the p-value to distinguish statistical significance from practical importance.